Theorem uspgrushgr 29334

Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)

Assertion

Ref	Expression
uspgrushgr	⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Proof of Theorem uspgrushgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2761	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	isuspgr 29309	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4		ssrab2 4031	. . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})
5		f1ss 6761	. . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
6	4, 5	mpan2 701	. . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7	3, 6	biimtrdi 255	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
8	1, 2	isushgr 29218	. . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})))
9	7, 8	sylibrd 261	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph))
10	9	pm2.43i 52	1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)

Syntax hints:   → wi 4   ∈ wcel 2141  {crab 3413   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4552  {csn 4579   class class class wbr 5097  dom cdm 5643  –1-1→wf1 6512  ‘cfv 6515   ≤ cle 11210  2c2 12265  ♯chash 14336  Vtxcvtx 29153  iEdgciedg 29154  USHGraphcushgr 29214  USPGraphcuspgr 29305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fv 6523  df-ushgr 29216  df-uspgr 29307

This theorem is referenced by:  uspgrupgrushgr  29336  usgredgedg  29387  vtxdusgrfvedg  29648  1loopgrvd2  29660  isubgr3stgr  48557